Understanding a paragraph about object detection with two objects
If I understood that post right (I just skimmed through, so it's possible I missed some details), they are using several predictors on the inputs with several rectangles. This basically means separate, not shared, weights to detect each rectangle.

It's fairly likely that these neurons will adjust to certain areas on the input image. As far as the neurons are concerned, they are "punished" if they detect the wrong rectangle, as if the other rectangle is the noise that the network should learn to ignore. As a result, the "left" predictor learns to fire more actively when it detects a rectangle on the left, because there's high chance it's the correct one. Same for the "right" predictor.

When, all of a sudden, the training image swaps the targets, each predictor gets confused. Because now what they used to detect as noise becomes the ground truth and vice versa. It certainly hurts the performance and makes the network be sure the most, when the target is around the center. When the targets are flipped, the dataset becomes more consistent and less confusing for the network, it's no wonder that the accuracy goes up.